What about this?

#!/usr/bin/perl
use strict;
my ($guess,$num);

# get number
while(<STDIN>) {
  $guess = $num = $_+0; last if ($num>0);
}

# iterate until within 1e-13
# go further and you need some math module, I think :)
while (abs($num - $guess**2) > 1e-13) {
    $guess = (($num/$guess) + $guess)/2;
}

# print our guess and what Perl says
print "My guess of square root of $num is $guess\n";
print "Perl says it's ".$num**.5."\n";
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cLive ;-)

PS - golf version? :)

#!/usr/bin/perl
use strict;
my $n;

while(<STDIN>) { $n = $_+0; last if $n>0}

while (abs($n - $_**2) > 1e-13){ $_ = ($n/$_ + $_)/2}

print "Guess: $_\nperl: ".$n**.5."\n";
[download]